Maybe the branch should be extended; ......................... and south to Waterloo
At Waterloo a non-passenger link with the Waterloo & City should be included, for stock transfers. Both W&C and the extended Aldwych branch could use shorter versions of the same trains
Having spent two-thirds of the mornings of my working life commuting through Waterloo to the Holborn area, I've had plenty of time to daydream whilst queuing for the 501/521 bus (before multi-door entry was introduced with the Bendybuses the queues used to stretch onto the main concourse of the station). My favourite daydream was designing the Waterloo extension. Much simpler than that proposed in the 1960s,which had new platforms below the Northern/Bakerloo ones, mine made more efficient use of the W&C platforms.
The W&C approaches waterloo from the north east, along Stamford Street, and passes under the IMAX roundabout in a right-angled curve to arrive facing SE, athwart the main line platforms above.
Adding a trailing crossover on the approach to Waterloo would allow W&C trains to reverse in the current arrivals platform, allowing them to return to Bank without shunting through the depot. (This would improve turnround time and possibly save one unit).
The current departures platform could then have been used for an extension of the Aldwych branch, with a tunnel branching off straight on where the existing W&C line turns NE towards Bank.
By re-purposing the existing platforms, the only new infrastructure required would be the new running tunnels and the trailing crossover.
The Holborn/Waterloo route could have been single track, but twin tunnels exist on the existing section of the route and potential demand suggests at least a passing place at Aldwych would have been needed to allow two train operation, and probably double track throughout except at the termini to allow an operation similar to the W&C.
Throughput would be improved if Aldwych station itself did not re-open. It would in any case be problematic - given the volume of end-to-end traffic - no-one would be able to board at Aldwych! (Similar considerations apply to Blackfriars calls on the W&C, desirable as that would have been for me when my place of work moved to Blackfriars later in my career!)
The connection between the former departures platform (now the platform for Holborn) and the line to Bank could be retained to allow stock transfers between the W&C and the rest of the Tube network without shunting through the depot (removing the need for a huge crane and associated road closures every time stock needs to go in or out).
A common fleet for the two shuttles would be possible - although the existing depot would not be large enough for this larger fleet, some could be stabled at the north end of the Piccadilly line. Indeed, Waterloo depot might be dispensed with altogether, allowing extension of the platforms at Waterloo (although this would also require extensions at Bank (possibly by shortening the Travelator?).
(I really did spend a lot of time in that queue - then I discovered it was just as quick to walk (and £1000pa cheaper, since I could exchange my Travelcard for a pointy-to-point season), and later still the "Boris Bike" scheme reduced the trip from 30minutes to less than ten)